arXiv:hep-th/0106011 · Journal of High Energy Physics

Quantum Hall states on the cylinder as unitary matrix Chern-Simons theory

Alexios P. Polychronakos

Published Jun 29, 2001• 74 citations

arXiv HTML PDF

DOI: 10.1088/1126-6708/2001/06/070

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Hall Viscosity in the Chern-Simons Matrix Model of the Laughlin Fractional Quantum Hall States

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arXiv:hep-th/0106011 · Journal of High Energy Physics

Quantum Hall states on the cylinder as unitary matrix Chern-Simons theory

Alexios P. Polychronakos

Published Jun 29, 2001• 74 citations

arXiv HTML PDF

DOI: 10.1088/1126-6708/2001/06/070

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Hall Viscosity in the Chern-Simons Matrix Model of the Laughlin Fractional Quantum Hall States

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Fractional Quantum Hall Effect and Topological Order: Wavefunctions, Anyons, and Chern-Simons Field Theory

Abednour Laidi

Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.

Oct 12, 20260 citations

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Chern-Simons Theory and Non-Abelian Anyons in Fractional Quantum Hall States — E8 Intelligence Research

Andrew Stewart Caldin

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Oct 12, 20260 citations

FINDING: Chern-Simons theory at level

k

governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action

S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)

; level

k \in \mathbb{Z}

(integer QHE) or

k = p/q

(fractional QHE, e.g.,

k=1/3

for Laughlin state); anyon braiding phase

\theta = \pi/k

(abelian) or unitary matrix

U_{ab}

(non-abelian, e.g., Fibonacci anyons with

U \sim e^{i\pi/5}

); entanglement entropy

S_A = \alpha L - \gamma

, with topological entanglement entropy

\gamma = \ln \sqrt{D}

D = \sum_i d_i^2

(quantum dimensions); lattice Chern-Simons reproduces continuum in limit

a \to 0

with

k

quantized. CONNECTION: Level

k=5

gives Fibonacci anyons with quantum dimension

d = \phi = 1.618

(golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com